Year 11 General Matrices And Matrix Arithmetic

Matrix Multiplication And Power Of A Matrix

20 practice questions 2 video lessons Theory + worked examples

Theory

Matrix multiplication uses the row-by-column rule: to find entry $(A B)_{i j}$ , take row $i$ of $A$ , column $j$ of $B$ , multiply corresponding entries, and add. The product is defined only when $A$ 's columns equal $B$ 's rows. Powers like $A^{2}$ only make sense for square matrices. Matrix multiplication is not commutative.

Matrix multiplication uses a row-by-column rule, and only works when the dimensions "match up".

If $A$ is $m \times n$ and $B$ is $n \times p$ , then $A B$ is defined and has order $m \times p$ . The "inner" dimensions (the two $n$ s) must match; the "outer" dimensions give the order of the result.

Each entry of the product is found by taking the corresponding row of the left matrix and column of the right matrix, multiplying entries pairwise, and adding:

$(A B)_{i j} = \sum_{k} A_{i k} B_{k j}$

For a square matrix $A$ , the power $A^{2} = A \times A$ , and $A^{3} = A \times A \times A$ . Powers only make sense for square matrices, since $A A$ needs $A$ 's columns to equal $A$ 's rows.

The first diagram shows the dimension rule: inner dimensions must match. The second visualises the row-by-column rule using a single entry of the product.

For

A B

to be defined, columns of

A

must equal rows of

B

; outer dimensions give the result.

Top-left entry: pair row 1 of

A

with column 1 of

B

, multiply, add:

1 \cdot 5 + 2 \cdot 7 = 19

The key idea is a single rule plus a dimension check.

The row-by-column rule

If $A$ is $m \times n$ and $B$ is $n \times p$ , then $A B$ is $m \times p$ , with

$(A B)_{i j} = A_{i 1} B_{1 j} + A_{i 2} B_{2 j} + \dots + A_{i n} B_{n j}$

(A B)_{i j} = \sum_{k} A_{i k} B_{k j}

Powers of a square matrix

$A^{2} = A \times A, A^{3} = A \times A \times A$

Powers are only defined when $A$ is square — i.e. $n \times n$ .

Key properties

Property	Statement
Associative	$(A B) C = A (B C)$
Distributive	$A (B + C) = A B + A C$
Identity	$A I = I A = A$
Zero	$A 0 = 0 A = 0$
Not commutative	$A B \neq B A$ in general

A2 is not entry-squaring. A2 means A×A using the row-by-column rule. Squaring each individual entry of A gives a different (usually wrong) matrix.

How to compute $A B$

Check the dimensions. If $A$ is $m \times n$ and $B$ is $n \times p$ , the product is defined. The result has order $m \times p$ .
Set up a blank result matrix of order $m \times p$ .
For each entry $(A B)_{i j}$ , take row $i$ of $A$ and column $j$ of $B$ , multiply matching entries, and add.
Fill in every entry of the result.

How to compute $A^{2}$ for a square matrix

Write $A^{2} = A \times A$ .
Apply the row-by-column rule on the two copies of $A$ .
The result has the same order as $A$ .

How to find a missing entry in a product equation

Locate the entry of the product that depends on the unknown.
Write the row-by-column equation for that entry and substitute known values.
Solve the resulting linear equation for the unknown.

EXAMPLE 1 — STANDARD 2×2 PRODUCT

Find

A B

where

A = [\begin{matrix} 2 & - 1 \\ 3 & 4 \end{matrix}]

and

B = [\begin{matrix} - 1 & 2 \\ 5 & - 3 \end{matrix}]

SOLUTION

Both matrices are $2 \times 2$ , so $A B$ is also $2 \times 2$ . Apply the row-by-column rule four times.

Top-left	$=$	$(2) (- 1) + (- 1) (5) = - 7$
Top-right	$=$	$(2) (2) + (- 1) (- 3) = 7$
Bottom-left	$=$	$(3) (- 1) + (4) (5) = 17$
Bottom-right	$=$	$(3) (2) + (4) (- 3) = - 6$

$A B = [\begin{matrix} - 7 & 7 \\ 17 & - 6 \end{matrix}]$

A B = [\begin{matrix} - 7 & 7 \\ 17 & - 6 \end{matrix}]

EXAMPLE 2 — ROW × COLUMN

Find

A B

where

A = [\begin{matrix} 2 & 3 & 1 \end{matrix}]

and

B = [\begin{matrix} 4 \\ 5 \\ 6 \end{matrix}]

SOLUTION

$A$ is $1 \times 3$ and $B$ is $3 \times 1$ , so $A B$ is $1 \times 1$ — a single entry.

$A B$	$=$	$(2) (4) + (3) (5) + (1) (6)$
	$=$	$8 + 15 + 6$
	$=$	$[\begin{matrix} 29 \end{matrix}]$

Answer: $A B = [\begin{matrix} 29 \end{matrix}]$ .

A B = [29]

EXAMPLE 3 — POWER A²

Find

A^{2}

where

A = [\begin{matrix} 3 & 1 \\ 0 & 2 \end{matrix}]

SOLUTION

$A^{2} = A \times A$ . $A$ is square ( $2 \times 2$ ), so the product is defined.

Top-left	$=$	$(3) (3) + (1) (0) = 9$
Top-right	$=$	$(3) (1) + (1) (2) = 5$
Bottom-left	$=$	$(0) (3) + (2) (0) = 0$
Bottom-right	$=$	$(0) (1) + (2) (2) = 4$

$A^{2} = [\begin{matrix} 9 & 5 \\ 0 & 4 \end{matrix}]$

A^{2} = [\begin{matrix} 9 & 5 \\ 0 & 4 \end{matrix}]

EXAMPLE 4 — FIND A MISSING ENTRY

Find

x

such that

[\begin{matrix} 2 & x \\ 1 & 3 \end{matrix}] [\begin{matrix} 4 & 1 \\ 2 & 5 \end{matrix}] = [\begin{matrix} 14 & ? \\ 10 & 16 \end{matrix}]

SOLUTION

The top-left entry of the product comes from row 1 of the left matrix and column 1 of the right matrix.

$(2) (4) + (x) (2)$	$=$	$14$
$8 + 2 x$	$=$	$14$
$2 x$	$=$	$6$
$x$	$=$	$3$

Answer: $x = 3$ .

x = 3

Common pitfalls

Matrix multiplication is not commutative. In general

A B \neq B A

. Even when both products are defined, they usually give different results — and often only one of them is defined.

Check the orders before starting. A

3 \times 2

matrix cannot multiply another

3 \times 2

(since

2 \neq 3

). A

3 \times 2

can multiply a

2 \times 4

to give a

3 \times 4

result.

$A^{2}$ is not entry-by-entry squaring.

A^{2} = A \times A

using the row-by-column rule. The matrix you get this way is very different from the one you'd get by squaring each entry separately.

Powers are square-only.

A^{2}

only makes sense when

A

is a square matrix. If

A

2 \times 3

, then

A \times A

is undefined (inner dimensions

3 \neq 2

Row × column, not column × row. The rule pairs the

i

-th row of the LEFT matrix with the

j

-th column of the RIGHT matrix. Mixing up which side is which gives the wrong answer.

Frequently asked questions

When is the product AB defined?

Only when the number of columns of A equals the number of rows of B. If A is m by n and B is n by p, then AB is defined and has order m by p. If the inner dimensions do not match, the product does not exist.

How do I compute each entry of AB?

Each entry (AB) subscript ij is found by taking the i-th row of A and the j-th column of B, multiplying matching entries together, and adding the results. This is called the row-by-column rule.

Is matrix multiplication commutative?

No. In general AB does not equal BA. Sometimes only one of the two is even defined. Always do the multiplication in the order asked.

What is A squared?

A squared means A times A using the row-by-column rule. It is only defined for square matrices, because you need A's columns to match A's rows. A squared is NOT the same as squaring each entry.

What is the identity matrix's role in multiplication?

Multiplying any matrix A by the identity matrix I of the right size leaves A unchanged. That is, AI equals IA equals A. The identity plays the role of the number 1 in ordinary multiplication.

How do I find a missing entry in a matrix product?

Use the row-by-column rule to write the equation for the entry that contains the unknown. That gives one linear equation in one variable, which you can solve directly.

Video Lessons

Practice Questions

20 questions available.

Practice Questions

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Problem-Solving And Modelling With Matrices

Matrix Multiplication And Power Of A Matrix

📖 Theory

The row-by-column rule

Powers of a square matrix

Key properties

How to compute AB

How to compute A2 for a square matrix

How to find a missing entry in a product equation

Common pitfalls

Frequently asked questions

🎬 Video Lessons

✏️ Practice Questions

Theory

How to compute $A B$

How to compute $A^{2}$ for a square matrix

Video Lessons

Practice Questions